Periodica Mathematica Hungarica

, Volume 51, Issue 2, pp 75–107

# Modular constructions of pseudorandom binary sequences with composite moduli

• Joël Rivat
• András Sárközy
Article

## Summary

Recently, Goubin, Mauduit, Rivat and Sárközy have given three constructions for large families of binary sequences. In each of these constructions the sequence is defined by modulo <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[\$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>p\$ congruences where \$p\$ is a prime number. In this paper the three constructions are extended to the case when the modulus is of the form \$pq\$ where \$p\$, \$q\$ are two distinct primes not far apart (note that the well-known Blum-Blum-Shub and RSA constructions for pseudorandom sequences are also of this type). It is shown that these modulo \$pq\$ constructions also have certain strong pseudorandom properties but, e.g., the (``long range'') correlation of order \$4\$ is large (similar phenomenon may occur in other modulo \$pq\$ constructions as well).

correlation binary sequence pseudo-random additive character